Properness and Widely Linear Processing of Quaternion Random Vectors

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Executive Summary

In this paper, the second-order circularity of quaternion random vectors is analyzed. Unlike the case of complex vectors, there exist three different kinds of quaternion properness, which are based on the vanishing of three different complementary covariance matrices. The different kinds of properness have direct implications on the Cayley - Dickson representation of the quaternion vector, and also on several well-known multivariate statistical analysis methods. In particular, the quaternion extensions of the Partial Least Squares (PLS), Multiple Linear Regression (MLR) and Canonical Correlation Analysis (CCA) techniques are analyzed, showing that, in general, the optimal linear processing is full-widely linear.

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